Hopscotch method: The numerical solution of the Frank-Kamenetskii partial differential equation

Keywords: Hopscotch scheme Thermal explosion Nonlinear source term Linear stability analysis a b s t r a c t Numerical solutions to the Frank-Kamenetskii partial differential equation modelling a thermal explosion in a cylindrical vessel are obtained using the hopscotch scheme. We observe that a nonlinear source term in the equation leads to numerical difficulty and hence adjust the scheme to accommodate such a term. Numerical solutions obtained via MATLAB, MATHEMATICA and the Crank–Nicolson implicit scheme are employed as a means of comparison. To gain insight into the accuracy of the hopscotch scheme the solution is compared to a power series solution obtained via the Lie group method. The numerical solution is also observed to converge to a well-known steady state solution. A linear stability analysis is performed to validate the stability of the results obtained. In this paper we consider numerical solutions to the Frank-Kamenetskii [12] partial differential equation which is capable of modelling a thermal explosion in a cylindrical vessel. The heat balance equation, neglecting reactant consumption, is given by c @T @t ¼ jr 2 T þ rQA exp À E RT : ð1Þ In this formulation, c is the thermal capacity, j the thermal conductivity, r the density, Q the heat of the reaction, A the frequency factor, E the energy of activation of the chemical reaction, R the universal gas constant and T the gas temperature. For this kind of chemical process it must be noted that c, j, r and Q would likely evolve as the heat increases. However it has been noted in the work done by Frank-Kamenetskii [12] that Eq. (1) is a reasonable model for the physical process in question and able to provide us with reasonable insight into the temperature distribution. A thermal explosion occurs when the heat generated by a chemical reaction is far greater than the heat lost to the walls of the vessel in which the reaction is taking place. The increase in the heat generated by the reaction increases the rate of the chemical reaction exponentially following the Arrhenius equation until a thermal explosion occurs. Eq. (1) was derived by Frank-Kamenetskii [12]. Further work was done by Steggerda [33] on Frank-Kamenetskii's original criterion for a thermal explosion showing that a more detailed consideration of the situation is possible. Rice [31] has applied the theory developed by Frank-Kamenetskii for thermal explosions in which heat is removed by …